Math 104
Fall 2009
Homework 2: solution set
We use the notations below to ease readability. Matrices are bold capital, vectors are bold lowercase
and scalars or entries are not bold. For instance,
A
is a matrix and
a
ij
(sometimes,
A
(
i,j
)) its (
i,j
)th
entry. Likewise
x
is a vector and
x
j
its
j
th component. The linear span generated by a group of vecotors
v
1
,
v
2
,...,
v
n
is denoted by span(
v
1
,
v
2
,...,
v
n
).
Problem 1
Since
(
AB
)(
B

1
A

1
) =
A
(
BB

1
)
A

1
=
AIA

1
=
AA

1
=
I
,
we have that
AB
is invertible and the inverse of
AB
is
B

1
A

1
.
Problem 2
Suppose that
R
=
r
11
···
r
1
m
.
.
.
.
.
.
.
.
.
r
m
1
···
r
mm
= [
r
1
,
···
,
r
m
]
is a nonsingular uppertriangular matrix, which means
{
r
1
,
···
,
r
m
}
forms a basis of
C
m
, and
r
ij
= 0
for
i > j
. Suppose
e
j
is the canonical unit vector with 1 in the
j
th entry and zeros elsewhere. Then
r
j
=
∑
j
i
=1
r
ij
e
i
, which implies
r
j
∈
span(
e
1
,...,
e
j
) and so span(
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 Fall '09
 Linear Algebra, Vectors, Matrices, Scalar, 1 m, ej, th column

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